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The Brownian motion is one of the most interesting and useful of all Stochastic Processes. It has an enormous range of applications ranging from physics (Einstein) to finance (starting with Bachelier).

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We will show that there is a family of n- variate polynomials of degree d = O(log^2 n), which can be computed by linear sized homogeneous depth-5 arithmetic circuits, where as any homogeneous depth-4 circuit computing it must have size at least n^

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Cheeger inequalities in spectral graph theory help to comment on the approximate connectivity or expansion of a graph (a combinatorial property) from the eigenvalues of the adjacency matrix of the graph (an algebraic property).

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The financial crisis of 2007‐2009 has given way to the sovereign debt crisis of 2010‐2012, yet many of the banking issues remain the same. We discuss a method to estimate the capital that a financial firm would need to rais